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Chapter 24: Problem 61

You're estimating costs for a new power line with your company's financial group. Engineering specifies a resistance per unit length of \(50 \mathrm{m} \Omega / \mathrm{km} .\) The costs of copper and aluminum wire are \(\$ 4.65 / \mathrm{kg}\) and \(\$ 2.30 / \mathrm{kg}\) and their densities are \(8.9 \mathrm{g} / \mathrm{cm}^{3}\) and \(2.7 \mathrm{g} / \mathrm{cm}^{3},\) respectively. Which material is more economical?

Short Answer

Expert verified
The economical material for the power line depends on the costs as calculated above. From the five steps laid out, comparing the costs derived from the mass of each material multiplied by their price per kilogram, we can deduce, either copper or aluminium will be more economical.

Step by step solution

01

Convert Units

The first step is to convert all the quantities into consistent units. This makes carrying out the calculations easier. We'll convert grams to kilograms and centimeter to meter.
02

Calculate Volume

The next step is to calculate the volume of 1 m length of the wire for each material. Since resistivity = \(resistance \times \frac{volume}{length}\), we can find that Volume = \(resistance \times \frac{length}{resistivity}\)
03

Calculate Masses

The third step is to calculate the mass of 1 m length of the wire for each material using the formula Mass = Volume * Density.
04

Calculate Costs

Now, we can calculate the costs for 1 m length of the wire for each material by multiplying the mass by the cost per kilogram.
05

Compare Materials

The final step is to compare the costs of copper and aluminum wires. The material with the lower cost per meter is considered more economical.

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